Curve registration in functional data analysis with informatively censored event-times

Curve Registration is a technique for aligning a set of curves whose time scale is observed subject to random error. In this dissertation, a general approach to Curve Registration for longitudinal and functional data, in the possible presence of informative dropout and time-varying treatments, is developed. A new method is developed for fitting the Semiparametric Nonlinear Mixed Effects Model (SNMM) where a B-spline basis expansion is used to estimate the common shape function. By using a smoothing spline to estimate the common shape function, the existing approaches to estimation and inference in this framework do not estimate the model parameters from a unified likelihood-based optimization criterion and instead use a backfitting approach that iterates between two mixed effects models. Such an iterative algorithm will not be guaranteed to converge, and because the variability in each of the two models is not properly accounted for, statistical inferences based on this approach may not be valid. Instead, a B-spline basis expansion is used in place of the smoothing spline which unifies estimation of all parameters within the same likelihood. Convergence is guaranteed, and likelihood-based statistical inferences and model selection will be valid. Computationally, the algorithm is simplified in comparison to the smoothing spline approach because the dimension of integration needed to compute the log-likelihood is typically small. Therefore, a more accurate numerical integration scheme based on Adaptive Gaussian Quadrature is implemented. The SNMM is extended to the shared parameter framework to enable joint modeling of the longitudinal trajectories and informatively censored event-times. Time-varying treatments are also accommodated through another extension to the branching curve problem. The methods developed in this dissertation are applied to a Women\u27s Health study involving women attempting a vaginal birth after cesarean (VBAC). The results of fitting the SNMM and its extensions are used to characterize the average progression of labor, to determine whether cases of uterine rupture tended to have longer delivery times, on average, than healthy controls, to model the effect of oxytocin, a labor inducing agent, on the average labor progressions